Radial index of Laguerre-Gaussian modes in high-order-harmonic generation

High-order-harmonic generation (HHG) is a tabletop and tunable source of extreme ultraviolet (XUV) light. Its flexibility was lately evidenced by the production of Laguerre-Gaussian (LG) modes in the XUV with a known azimuthal index. Here we investigate the role of the radial index of LG modes in HHG. We show both numerically and experimentally that the mode content of the emitted XUV radiation can be tuned by controlling the weight of the different quantum trajectories involved in the process. The appearance of high-order radial modes is finally linked to the atomic dipole phase of HHG. These results extend the capabilities of shaping the spatial mode of ultrashort XUV pulses of light.

Figure 1

Contributions of different quantum trajectories in LG-HHG. The intensity of the 21st harmonic in argon is computed in the far field for (a) $z_0<0$, (b) $z_0=0$, and (c) $z_0>0$, where $z_0$ is the position of the laser focus compared to the gas jet. (d) and (e) show the profile in $z_0=0$ when only the short or long trajectory contribution is selected.

Figure 2

Measurement of the harmonic intensity profile as a function of the focus position. (a) Experimental setup for LG-HHG and measurement of the 2D spatial profile of several harmonics. The acronyms used for the optical elements are defined in the text. (b) Intensity profiles of harmonics 11–15 when the generating lens is translated over 1.2 cm. The intensity of each harmonic is normalized to 1. $z=0$ is known from previous experiments done with Gaussian drivers on this beamline.

Figure 3

Modal analysis of quantum trajectories. Left: Coefficients of the decomposition of the electric field of harmonic 19 onto the Laguerre-Gaussian basis for the short and long trajectories. Coefficients for other values of ${\ell }_{19}$ are null. Right: Intensity of harmonic 19 for the short and long trajectories upon numerical propagation over 200 mm.

Figure 4

Mode content as a function of the driving intensity. Coefficients of the decomposition of harmonic 19 when the infrared peak intensity is varied from to 1 to $2.4\times {10}^{14}$ W/cm.